Question

Using the chart below showing the age and salary of each NBA Starting Point Guard, Let x = age and y = salary

1. Is there a correlation between the two variables? Explain.

2. Find the equation of the regression line THEN use it to predict the salary of a 42-year-old point guard.

Age	Salary
30	41358814
31	40231758
29	39200000
22	33930000
23	33930000
31	30521115
29	27977689
29	26361000
33	17000450
29	16000000
27	14000000
28	42782880
34	39932648
29	35197650
27	34122650
23	29331375
26	21250000
29	17500000
25	17500000
25	12999975
27	12500000
30	12000000
28	9607500
31	7250000
23	6655113
25	4372072
22	3627677
26	1892139
22	1589844
19	1532398

Answer 1

Solution

Note:

Solution are obtained directly using Excel functions as indicated therein.

However, the Back-up Theory is given at the end to get a deeper insight into the subject.

Now, to work out the solution,

1.

Correlation coefficient, r = 0.4236 [using Excel function: Statistical CORREL] Answer 1

2.

The equation of the estimated regression line is: y_hat = a + bx.

Where

a = y-intercept = -22464268.84 [using Excel function: Statistical INTERCEPT]

b = slope = 1608476.3696 [using Excel function: Statistical SLOPE]

Substituting the values,

equation of the estimated regression line is: y_hat = 1608476.3696x - 22464268.84 Answer 2

Predicted salary of a 42 year-old person is: 45091738.69 Answer 3

[obtained by substituting x = 42 in Answer 2]

Back-up Theory

The linear regression model: Y = β₀ + β₁X + ε, ………………………………..........................................................………..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y_cap = β_0cap + β_1capX, …………….................................................……….(2)

where β_1cap = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) and β_0cap = Ybar – β_1cap.Xbar..………………...........…………….…..(3)

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i …………………………….........................................................…………….……….….(4)

Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i …………………………….........................................................…………….……….….(5)

Sxx = Σ(i = 1 to n)(x_i – Xbar)² ………………………............................................................………………………..…………....(6)

Syy = Σ(i = 1 to n)(y_i – Ybar)² ……………………….............................................................……………………..………………(7)

Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ……………..........................................................………………………………….(8)

Correlation coefficient, r = Sxy/sqrt(Sxx. Syy) ……........................................................……………………………….. (9)

DONE